Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Annales Polonici Mathematici / All issues

Abstract

We consider the Dirichlet problem for the complex Monge–Ampère equation in a bounded strongly hyperconvex Lipschitz domain in $\mathbb C^n$. We first give a sharp estimate on the modulus of continuity of the solution when the boundary data is continuous and the right hand side has a continuous density. Then we consider the case when the boundary value function is $\mathcal {C}^{1,1}$ and the right hand side has a density in $L^p(\varOmega )$ for some $p>1$, and prove the Hölder continuity of the solution.